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A "harmonious graph" is one that has a harmonious labeling. Odd cycles are harmonious, as are Petersen graphs. It is conjectured that trees are all harmonious if one vertex label is allowed to be reused. [8] The seven-page book graph K 1,7 × K 2 provides an example of a graph that is not harmonious. [9]
Every graph has a harmonious coloring, since it suffices to assign every vertex a distinct color; thus χ H (G) ≤ | V(G) |. There trivially exist graphs G with χ H (G) > χ(G) (where χ is the chromatic number); one example is any path of length > 2, which can be 2-colored but has no harmonious coloring with 2 colors. Some properties of χ H ...
Optimal (span-5) radio coloring of a 6-cycle. In graph theory, a branch of mathematics, a radio coloring of an undirected graph is a form of graph coloring in which one assigns positive integer labels to the graphs such that the labels of adjacent vertices differ by at least two, and the labels of vertices at distance two from each other differ by at least one.
The clique-width of a graph G is the minimum number of distinct labels needed to construct G by operations that create a labeled vertex, form the disjoint union of two labeled graphs, add an edge connecting all pairs of vertices with given labels, or relabel all vertices with a given label. The graphs of clique-width at most 2 are exactly the ...
A graceful labeling. Vertex labels are in black, edge labels in red.. In graph theory, a graceful labeling of a graph with m edges is a labeling of its vertices with some subset of the integers from 0 to m inclusive, such that no two vertices share a label, and each edge is uniquely identified by the absolute difference between its endpoints, such that this magnitude lies between 1 and m ...
For the example graph, P(G, t) = t(t − 1) 2 (t − 2), and indeed P(G, 4) = 72. The chromatic polynomial includes more information about the colorability of G than does the chromatic number. Indeed, χ is the smallest positive integer that is not a zero of the chromatic polynomial χ(G) = min{k : P(G, k) > 0}.
A coloring of a given graph is distinguishing for that graph if and only if it is distinguishing for the complement graph. Therefore, every graph has the same distinguishing number as its complement. [2] For every graph G, the distinguishing number of G is at most proportional to the logarithm of the number of automorphisms of G.
An r-tuple incidence k-coloring of a graph G is the assignment of r colors to each incidence of graph G from a set of k colors such that the adjacent incidences are given disjoint sets of colors. [14] By definition, it is obvious that 1-tuple incidence k-coloring is an incidence k-coloring too.